Base 15: Cycles Sequences: All Cycle Counts

Blog: The First Pixel: Kaprekar's Constant 6174

Summary
[All Digit Sets]
[Odd Digit Sets]
[Even Digit Sets]
Final Numbers
[Fixed Points]
Cycles Lengths
[All Digit Sets]
[Odd Digit Sets]
[Even Digit Sets]
Iteration Frequencies
[Cycle Start]
[Full Cycle]
Maximum Iteration Sequences
[Cycle Start]
[Full Cycle]
Cycles Sequences
[Maximum Length Cycle Count]
[All Cycle Counts] — [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [Base 15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178] [179] [180] [181] [182] [183] [184] [185] [186] [187] [188] [189] [190] [191] [192] [193] [194] [195] [196] [197] [198] [199] [200] [201] [202] [203] [204] [205] [206] [207] [208] [209] [210] [211] [212] [213] [214] [215] [216] [217] [218] [219] [220] [221] [222] [223] [224] [225] [226] [227] [228] [229] [230] [231] [232] [233] [234] [235] [236] [237] [238] [239] [240] [241] [242] [243] [244] [245] [246] [247] [248] [249] [250] [251] [252] [253] [254] [255]

Radix Character Encoding
0	1	2	3	4	5	6	7	8	9	A	B	C	D	E

Color Key
Orange	Fixed Point	number that becomes itself after one iteration
Green	Final Numbers	all numbers eventually reached, from all cycles
Purple	Cycle Start	iterations include entry into cycle, assuming it repeats
Red	Full Cycle	iterations include entire cycle, proving it repeats
Blue	Start Numbers	subset (optimization) of all numbers; minimal for full coverage
Gray	No Results	any calculation that has no results

Digits	# Full Cycles (excluding zero)	Frequency	Cycle Length	Full Cycles (excluding zero) (bold = exactly one cycle [excluding zero])
1	0 cycles	-	-	-
2	0 cycles	-	-	-
3	1 cycle	665×	(max) 2 nodes	⮎ 6E8 → 7E7 ⮌
4	7 cycles	658×	(max) 4 nodes	⮎ 30DC → D852 → B2B4 → 9676 ⮌
		609×	(max) 4 nodes	⮎ 70D8 → D0D2 → DA32 → B674 ⮌
		557×	(max) 4 nodes	⮎ 50DA → D492 → B494 → 7498 ⮌
		541×	(max) 4 nodes	⮎ 72B8 → 90D6 → D2B2 → B854 ⮌
		343×	2 nodes	⮎ 52BA → 9496 ⮌
		226×	2 nodes	⮎ B0D4 → D672 ⮌
		111×	(fixed) 1 node	⮎ 92B6 ⮌
5	1 cycle	11,613×	(fixed) (max) 1 node	⮎ A4E95 ⮌
6	2 cycles	31,322×	(max) 9 nodes	⮎ 41EECB → DA0D42 → DB5832 → B82B64 → 971C76 → B2EEB4 → C9EE43 → BA2B44 → 975876 ⮌
6	2 cycles	7,423×	3 nodes	⮎ 960D86 → D31CB2 → CA7643 ⮌
7	2 cycles	91,802×	(max) 10 nodes	⮎ B70ED74 → E93EA51 → DB4E932 → CA6E743 → B83EA64 → B73EA74 → B72EB74 → C73EA73 → B92EB54 → C75E873 ⮌
7	2 cycles	24,463×	8 nodes	⮎ A62EB85 → C63EA83 → B93EA54 → B74E974 → A71EC75 → D72EB72 → CB3EA33 → B97E654 ⮌
8	4 cycles	159,335×	(max) 22 nodes	⮎ 861EEC87 → D84EE962 → CA70D743 → D95EE852 → C980D653 → D941CA52 → CA63A843 → 9761C876 → B31EECB4 → DB7EE632 → CB73A733 → 986EE766 → 8830DB67 → D820DC62 → DBA1C432 → CA858643 → 9631CB86 → B831CB64 → B871C764 → B71EEC74 → DA53A942 → B763A874 ⮌
		136,895×	4 nodes	⮎ A940DA55 → D653A982 → B741CA74 → B75EE874 ⮌
		21,587×	3 nodes	⮎ 9660D886 → D321CCB2 → CAA76443 ⮌
		1,938×	(fixed) 1 node	⮎ A94EE955 ⮌
9	3 cycles	691,035×	(max) 8 nodes	⮎ A742EBA75 → C752EB973 → C961EC853 → D972EB752 → CB61EC833 → D994E9552 → C943EAA53 → B964E9854 ⮌
		112,357×	6 nodes	⮎ B862EB864 → C751EC973 → D971EC752 → DB71EC732 → DB93EA532 → CA84E9643 ⮌
		13,783×	(fixed) 1 node	⮎ C962EB853 ⮌
10	3 cycles	1,950,128×	(max) 6 nodes	⮎ B761EEC874 → DA640DA842 → DB661C8832 → CA821CC643 → BA961C8544 → B7641CA874 ⮌
		10,344×	3 nodes	⮎ 96660D8886 → D3221CCCB2 → CAAA764443 ⮌
		769×	(fixed) 1 node	⮎ AA54EE9945 ⮌
11	4 cycles	2,855,932×	4 nodes	⮎ C9431ECBA53 → D9974E97552 → C9441ECAA53 → D9864E98652 ⮌
		1,048,735×	9 nodes	⮎ B9530EDB954 → EA763EA8741 → DB640EDA832 → EBA73EA7431 → DB862EB8632 → CB851EC9633 → D9962EB8552 → CB642EBA833 → C9873EA7653 ⮌
		551,878×	(max) 11 nodes	⮎ B9641ECA854 → D8752EB9762 → CB630EDB833 → EA984E96541 → DA642EBA842 → CB763EA8733 → B9840EDA654 → E9752EB9751 → DC641ECA822 → DBA83EA6432 → CA863EA8643 ⮌
		840×	(fixed) 1 node	⮎ A994EEE9555 ⮌
12	2 cycles	9,633,826×	(max) 19 nodes	⮎ 764321CCBA88 → BA8620DC8644 → DA7621CC8742 → CAA61EEC8443 → DB8841CA6632 → CA8621CC8643 → BA9621CC8544 → BA7641CA8744 → B7661EEC8874 → DA6510DD9842 → DCB641CA8322 → CAA861C86443 → B96621CC8854 → BA7421CCA744 → BA765EE88744 → AA6410DDA845 → DC6651C98822 → CAA421CCA443 → BA9665888544 ⮌
12	2 cycles	23,859×	3 nodes	⮎ 966660D88886 → D32221CCCCB2 → CAAAA7644443 ⮌
13	2 cycles	20,057,505×	(max) 4 nodes	⮎ CB8641ECA8633 → D99741ECA7552 → DB8541ECA9632 → DB9752EB97532 ⮌
13	2 cycles	780×	(fixed) 1 node	⮎ AA954EEE99545 ⮌
14	4 cycles	36,952,282×	6 nodes	⮎ BA76421CCA8744 → BA7661EEC88744 → DA85410DDA9642 → DCB6641CA88322 → CAA8621CC86443 → BA96621CC88544 ⮌
		3,143,393×	(max) 7 nodes	⮎ 9666321CCB8886 → BA83221CCCB644 → BAA8761C876444 → B76621EECC8874 → DC86510DD98622 → DCBA421CCA4322 → CAAA8658864443 ⮌
		19,770×	3 nodes	⮎ 9666660D888886 → D322221CCCCCB2 → CAAAAA76444443 ⮌
		1,140×	(fixed) 1 node	⮎ A9994EEEE95555 ⮌
15	7 cycles	69,470,068×	(max) 33 nodes	⮎ CB75430EDBA9733 → EA98751EC976541 → DD85430EDBA9612 → ECBA752EB974321 → DCA8751EC976422 → DBA8530EDB96432 → EBA8752EB976431 → DC87530EDB97622 → EBB9630EDB85331 → EDA8862EB866411 → DDB7421ECCA7312 → DCBA972EB754322 → CBA8751EC976433 → D997530EDB97552 → EBA6441ECAA8431 → DD97663EA887512 → DBA5320EDCB9432 → EBB9874E9765331 → DB87430EDBA7632 → EBA8740EDA76431 → EDA7640EDA87411 → EDC9640EDA85211 → EDCB852EB963211 → DDBA862EB864312 → DBB8741ECA76332 → DB98740EDA76532 → EBA7530EDB97431 → EDA8751EC976411 → DDC8530EDB96212 → ECBB962EB853321 → DCA8862EB866422 → CBA7421ECCA7433 → DA99762EB875542 ⮌
		2,564,718×	4 nodes	⮎ CB88630EDB86633 → EA98521ECC96541 → DDA8542EBA96412 → DBB7652EB987332 ⮌
		2,324,238×	6 nodes	⮎ CB87530EDB97633 → EA98630EDB86541 → EDA7531ECB97411 → DDC9751EC975212 → DCBA741ECA74322 → DBA9762EB875432 ⮌
		1,169,621×	(fixed) 1 node	⮎ DB97541ECA97532 ⮌
		1,098,759×	2 nodes	⮎ DB97531ECB97532 → DB98641ECA86532 ⮌
		930,561×	(fixed) 1 node	⮎ DB98631ECB86532 ⮌
		780×	(fixed) 1 node	⮎ AAA554EEE999445 ⮌
16	6 cycles	143,459,304×	(max) 6 nodes	⮎ BA76661EEC888744 → DA854210DDCA9642 → DCBA6641CA884322 → CAA86621CC886443 → BA966221CCC88544 → BAA76421CCA87444 ⮌
		1,654,002×	5 nodes	⮎ B986621EECC88654 → DC87531EECB97622 → DCB97520DC975322 → DBAA843EEAA64432 → CBA76641CA887433 ⮌
		180,148×	2 nodes	⮎ CB96632EEBB88533 → CB98652EEB986533 ⮌
		98,636×	(fixed) 1 node	⮎ CC98642EEBA86523 ⮌
		29,490×	3 nodes	⮎ 96666660D8888886 → D3222221CCCCCCB2 → CAAAAAA764444443 ⮌
		1,080×	(fixed) 1 node	⮎ AA9954EEEE995545 ⮌
17	5 cycles	199,772,496×	5 nodes	⮎ DC886430EDBA86622 → EBB97421ECCA75331 → DDA98751EC9765412 → DCB85430EDBA96322 → EBB98752EB9765331 ⮌
		48,110,013×	(max) 11 nodes	⮎ CB876531ECB987633 → D9986320EDCB86552 → EBB96431ECBA85331 → DD988752EB9766512 → DBB64320EDCBA8332 → EBB98873EA7665331 → DB885320EDCB96632 → EBB98631ECB865331 → DD988631ECB866512 → DCB96321ECCB85322 → DBAA9862EB8654432 ⮌
		15,650,119×	5 nodes	⮎ DBB75430EDBA97332 → EBA88751EC9766431 → DD975320EDCB97512 → ECBB9641ECA853321 → DDA98752EB9765412 ⮌
		1,648,442×	4 nodes	⮎ CB886320EDCB86633 → EB998521ECC965531 → DDA96442EBAA85412 → DBB76652EB9887332 ⮌
		1,440×	(fixed) 1 node	⮎ A99994EEEEE955555 ⮌
18	5 cycles	337,876,918×	6 nodes	⮎ BA766661EEC8888744 → DA8542210DDCCA9642 → DCBAA6641CA8844322 → CAA866621CC8886443 → BA9662221CCCC88544 → BAAA76421CCA874444 ⮌
		133,351,106×	(max) 10 nodes	⮎ AA7542210DDCCA9745 → DCAA6653EEA9884422 → CCA86542EEBA986423 → CC987542EEBA976523 → CC9865320DCB986523 → DAA984421CCAA65442 → CAA666641CA8888443 → B96662221CCCC88854 → BAAA74221CCCA74444 → BAA76665EE88887444 ⮌
		162,793×	(fixed) 1 node	⮎ CC996632EEBB885523 ⮌
		43,688×	3 nodes	⮎ 966666660D88888886 → D32222221CCCCCCCB2 → CAAAAAAA7644444443 ⮌
		1,080×	(fixed) 1 node	⮎ AAA9554EEEE9995445 ⮌
19	4 cycles	526,821,328×	5 nodes	⮎ DCB865430EDBA986322 → EBB987531ECB9765331 → DD9886430EDBA866512 → ECBA75321ECCB974321 → DDAA98751EC97654412 ⮌
		186,981,917×	(max) 18 nodes	⮎ CB8754320EDCBA97633 → EB9987530EDB9765531 → EDA864430EDBAA86411 → EDCA76641ECA8874211 → DDCA86420EDCA864212 → ECBBA8641ECA8643321 → DDA987641ECA8765412 → DCB865320EDCB986322 → EBBA98631ECB8654331 → DD9887531ECB9766512 → DCB964320EDCBA85322 → EBBA98752EB97654331 → DC8875430EDBA976622 → EBB975320EDCB975331 → EDB988641ECA8665311 → DDC975321ECCB975212 → DCBAA9641ECA8544322 → DBA976652EB98875432 ⮌
		105,004,560×	8 nodes	⮎ DB9875430EDBA976532 → EBA875430EDBA976431 → EDA876530EDB9876411 → EDCA75320EDCB974211 → EDCBA9751EC97543211 → DDCA97541ECA9754212 → DCBA86541ECA9864322 → DBA976531ECB9875432 ⮌
		1,380×	(fixed) 1 node	⮎ AA99954EEEEE9955545 ⮌
20	5 cycles	1,340,271,218×	(max) 27 nodes	⮎ BAA7664221CCCA887444 → BAA766621EECC8887444 → DC88744210DDCAA76622 → DCBAA6621EECC8844322 → DCBA98743EEAA7654322 → CCA9765420DCA9875423 → DAA966441EECAA885442 → DC9866542EEBA9886522 → CCB865331EECBB986323 → DC9998652EEB98655522 → CCB764432EEBBAA87323 → CC99877430DBA7765523 → DA986441EEEECAA86542 → DCAA974420DCAA754422 → DBAA66663EEA88884432 → CBA764421EECCAA87433 → DC99876430DBA8765522 → DBA864421EECCAA86432 → DCB987643EEAA8765322 → CCA9754310DDBA975423 → DCA986643EEAA8865422 → CCA865431EECBA986423 → DC9987542EEBA9765522 → CCB8654320DCBA986323 → DAA9886421CCA8665442 → CAA6664221CCCA888443 → BAA9666221CCC8885444 ⮌
		51,642,345×	6 nodes	⮎ BA7666661EEC88888744 → DA85422210DDCCCA9642 → DCBAAA6641CA88444322 → CAA8666621CC88886443 → BA96622221CCCCC88544 → BAAAA76421CCA8744444 ⮌
		58,708×	3 nodes	⮎ 9666666660D888888886 → D322222221CCCCCCCCB2 → CAAAAAAAA76444444443 ⮌
		1,740×	(fixed) 1 node	⮎ A999994EEEEEE9555555 ⮌
		1,614×	(fixed) 1 node	⮎ AAAA5554EEEE99994445 ⮌
21	3 cycles	2,167,211,683×	8 nodes	⮎ DCB8654320EDCBA986322 → EBBA987531ECB97654331 → DD98875430EDBA9766512 → ECBA754320EDCBA974321 → EDBA987651EC987654311 → DDC9754320EDCBA975212 → ECBBA97541ECA97543321 → DDA9876541ECA98765412 ⮌
		152,746,322×	(max) 11 nodes	⮎ DC88643220EDCCBA86622 → EBBAA97421ECCA7544331 → DDA9876651EC988765412 → DCB8543220EDCCBA96322 → EBBAA98752EB976544331 → DC88765430EDBA9876622 → EBB9753220EDCCB975331 → EDBA988641ECA86654311 → DDC9765321ECCB9875212 → DCBAA96430EDBA8544322 → EBB9876652EB988765331 ⮌
		1,380×	(fixed) 1 node	⮎ AAA99554EEEEE99955445 ⮌
22	6 cycles	1,607,563,975×	6 nodes	⮎ BA76666661EEC888888744 → DA854222210DDCCCCA9642 → DCBAAAA6641CA884444322 → CAA86666621CC888886443 → BA966222221CCCCCC88544 → BAAAAA76421CCA87444444 ⮌
		1,606,809,997×	(max) 16 nodes	⮎ BAA76642221CCCCA887444 → BAAA766621EECC88874444 → DC887644210DDCAA876622 → DCBAA66221EECCC8844322 → DCBAA98743EEAA76544322 → CCA97665420DCA98875423 → DAA9664421EECCAA885442 → DCB8866542EEBA98866322 → CCB9764321EECCBA875323 → DCA99976420DCA87555422 → DBAA664441EECAAA884432 → DCA8766643EEAA88876422 → CCA86442210DDCCAA86423 → DCAAA966621CC888544422 → CAAA6664221CCCA8884443 → BAA96666221CCC88885444 ⮌
		539,859,820×	3 nodes	⮎ DCB98754310DDBA9765322 → DCBA886441EECAA8664322 → DCB9876441EECAA8765322 ⮌
		41,997,387×	6 nodes	⮎ CBBA976641EECA88754333 → DB997754310DDBA9775532 → DCB886443EEEEAAA866322 → CCBB985441EECAA9653323 → DC999775530DB997755522 → DBA844443EEEEAAAAA6432 ⮌
		64,326×	3 nodes	⮎ 96666666660D8888888886 → D3222222221CCCCCCCCCB2 → CAAAAAAAAA764444444443 ⮌
		1,680×	(fixed) 1 node	⮎ AA999954EEEEEE99555545 ⮌
23	7 cycles	3,669,557,099×	(fixed) 1 node	⮎ EDCB9875430EDBA97653211 ⮌
		2,437,040,518×	(max) 11 nodes	⮎ DCB86543220EDCCBA986322 → EBBAA987531ECB976544331 → DD988765430EDBA98766512 → ECBA7543220EDCCBA974321 → EDBAA987651EC9876544311 → DDC97654320EDCBA9875212 → ECBBA975430EDBA97543321 → EDB98876541ECA987665311 → DDC97543220EDCCBA975212 → ECBBAA97541ECA975443321 → DDA98766541ECA988765412 ⮌
		338,852×	2 nodes	⮎ CBB9666332EEEBBB8885333 → CBB9886552EEEB998665333 ⮌
		83,856×	(fixed) 1 node	⮎ CBB9866532EEEBB98865333 ⮌
		63,040×	(fixed) 1 node	⮎ CCC9886442EEEBAA8665223 ⮌
		2,040×	(fixed) 1 node	⮎ A9999994EEEEEEE95555555 ⮌
		1,380×	(fixed) 1 node	⮎ AAAA95554EEEEE999954445 ⮌
24	9 cycles	7,402,593,606×	3 nodes	⮎ DCB987543210DDCBA9765322 → DCBAA886441EECAA86644322 → DCB98766441EECAA88765322 ⮌
		1,536,265,073×	3 nodes	⮎ DCB987654310DDBA98765322 → DCBA8864421EECCAA8664322 → DCBA9876441EECAA87654322 ⮌
		263,454,041×	(max) 16 nodes	⮎ AA7542222210DDCCCCCA9745 → DCAAAAA6653EEA9884444422 → CCA86666542EEBA988886423 → CC987543221EECCCBA976523 → DCAA99865320DCB986554422 → DBAA86644421CCAAA8864432 → CAA866666221CCC888886443 → BAA966222221CCCCCC885444 → BAAAAA766421CCA887444444 → BA766666621EECC888888744 → DC8854222210DDCCCCA96622 → DCBAAAAA6421CCA844444322 → CAAA86666661C88888864443 → B96662222221CCCCCCC88854 → BAAAAAA74221CCCA74444444 → BAA76666665EE88888887444 ⮌
		259,852,567×	3 nodes	⮎ DCB98764421EECCAA8765322 → DCBA98754310DDBA97654322 → DCBA8866441EECAA88664322 ⮌
		147,280,095×	10 nodes	⮎ CAA86666421EECCA88886443 → DC988644221EECCCAA866522 → DCBAA885431EECBA96644322 → DCB99776542EEBA987755322 → CCB97654320EEDCBA9875323 → ECB999765320DCB987555321 → ECAA8844441EECAAAA664421 → DDC88666643EEAA888866212 → DCBA8442221EECCCCAA64322 → DCBAAA987641CA8765444322 ⮌
		58,544,325×	6 nodes	⮎ BA766666661EEC8888888744 → DA8542222210DDCCCCCA9642 → DCBAAAAA6641CA8844444322 → CAA866666621CC8888886443 → BA9662222221CCCCCCC88544 → BAAAAAA76421CCA874444444 ⮌
		1,478,084×	(fixed) 1 node	⮎ DCB99866331EECBB88655322 ⮌
		84,614×	3 nodes	⮎ 966666666660D88888888886 → D32222222221CCCCCCCCCCB2 → CAAAAAAAAAA7644444444443 ⮌
		1,680×	(fixed) 1 node	⮎ AAA999554EEEEEE999555445 ⮌
25+	-	-	-	Not showing digit sets 25+ due to verbosity.